\chapter[Complex Analysis \\ \textnormal{\emph{Lectured in Lent \oldstylenums{2022} by \textsc{Prof.\ N.\ Wickramasekera}}}]{Complex Analysis}
\emph{\Large Lectured in Lent \oldstylenums{2022} by \textsc{Prof.\ N.\ Wickramasekera}}

Complex differentiation is a stronger notion than real differentiation.
Many functions that are differentiable as a function of two real variables are not complex differentiable, for example the complex conjugate function.
This stronger notion allows us to prove some surprising results.
It turns out that if a function is complex differentiable once in a neighbourhood of a point, then it is given by a convergent power series in some neighbourhood of that point.

Another interesting result is Cauchy's integral formula: if a function is complex differentiable in a neighbourhood around a point, one can evaluate the function at that point using a certain integral over any loop around that point.
A similar result can be used to obtain an arbitrary derivative of a function at a point by using a single integral.

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